Some inequalities on generalized Schur complements Original Research Article

This paper presents some inequalities on generalized Schur complements. Let A be an n×n (Hermitian) positive semidefinite matrix. Denote by A/α the generalized Schur complement of a principal submatrix indexed by a set α in A. Let A+ be the Moore–Penrose inverse of A and λ(A) be the eigenvalue vector of A. The main results of this paper are: 1. λ(A+(α′))greater-or-equal, slantedλ((A/α)+), where α′ is the complement of α in {1,2,…,n}. 2. λ(Ar/α)less-than-or-equals, slantλr(A/α) for any real number rgreater-or-equal, slanted1. 3. (C*AC)/αless-than-or-equals, slantC*/α A(α′) C/α for any matrix C of certain properties on partitioning.